In other words, we start with ideal \(\mathinner {|{+}\rangle }_{\rm gkp}\)  states and either apply the noise channel \({\cal N}_{\mathbf {Y}_{\rm gkp}}\)  or \({\cal N}_{\mathbf {Y}_{\rm p}}\)  with probabilities \(1-p_0\)  and \(p_0\) , respectively. Note that we know what the state is at each site, so we know which noise channel was applied. For GKP states, although this noise model does not capture the damping of peaks due to finite energy seen in eq:finiteEGKP, it captures the broadening of peaks that results from finite-energy effects [1]}, [2]}. Similarly, this method approximates the realistic momentum state well in the position basis but gives it a periodic structure in momentum space. These points can be viewed transparently through the Wigner picture. In both cases, the application of a noise channel renders the output states mixed.
